{"paper":{"title":"Three topological reducibilities for discontinuous functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Adam R. Day, Linda Brown Westrick, Rod Downey","submitted_at":"2019-06-18T14:19:41Z","abstract_excerpt":"We define a family of three related reducibilities, $\\leq_T$, $\\leq_{tt}$ and $\\leq_m$, for arbitrary functions $f,g:X\\rightarrow\\mathbb R$, where $X$ is a compact separable metric space. The $\\equiv_T$-equivalence classes mostly coincide with the proper Baire classes. We show that certain $\\alpha$-jump functions $j_\\alpha:2^\\omega\\rightarrow \\mathbb R$ are $\\leq_m$-minimal in their Baire class. Within the Baire 1 functions, we completely characterize the degree structure associated to $\\leq_{tt}$ and $\\leq_m$, finding an exact match to the $\\alpha$ hierarchy introduced by Bourgain and analyze"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07600","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}